# HW8 - 1. Construct a rectangle with its two side lengths...

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1. Construct a rectangle with its two side lengths generated independently from the Uniform (0 , 1) distribution. (a) Find the pdf of the area of this rectangle. (b) Find the pdf of the perimeter of this rectangle. (c) Find the pdf of the area of the largest square inscribed inside this rect- angle. 2. Let X 1 N ( μ 1 2 ) , X 2 N ( μ 2 2 ) be independent. Show that the conditional distribution of X 1 given X = X 1 + X 2 is normal and ﬁnd the mean and variance of this distributions (as a function of X ). [Hint: Use the same complete-the-square trick used in class to derive the distribution of X ] 3. Let X , Y have joint density given by f ( x,y ) = Γ( a + b + c ) Γ( a )Γ( b )Γ( c ) x a - 1 y b - 1 (1 - x - y ) c - 1 , x 0 ,y 0 ,x + y 1 , where a,b,c are positive integers. Find the marginal densities of V = X + Y and W = X/ ( X + Y ) . 4. Let X and Y be independent Uniform (0 , 1) random variables. Are X + Y and X - Y independent? Give arguments to support your answer. 5. Let a > 0 , b > 0 and suppose X Gamma ( a + b, 1) and Y Gamma ( a,b ) and X,Y are independent. Figure out the joint distribution of

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## This note was uploaded on 01/16/2011 for the course STAT 213 taught by Professor Ioannam during the Fall '09 term at Duke.

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HW8 - 1. Construct a rectangle with its two side lengths...

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